Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 9, Problem 9.16P
To determine
The bound state energies for hydrogen using WHB approximation in the form of Equation 9.52.
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Students have asked these similar questions
Problem 1:
This problem concerns a collection of N identical harmonic oscillators (perhaps an
Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf,
and so on.
a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving
1-x
the formula by long division. Prove it by first multiplying both sides of the
equation by (1 – x), and then thinking about the right-hand side of the resulting
expression.
b) Evaluate the partition function for a single harmonic oscillator. Use the result of
(a) to simplify your answer as much as possible.
c) Use E = -
дz
to find an expression for the average energy of a single oscillator.
z aB
Simplify as much as possible.
d) What is the total energy of the system of N oscillators at temperature T?
8.8 Calculate by direct integration the expectation values (r) and (1/r) of the radial position for
the ground state of hydrogen. Compare your results to the quoted expressions in Eq. (8.89)
and discuss your results. Did you expect that (1/r) # 1/(r)? Use your result for (1/r) to
find the expectation value of the kinetic energy of the ground state of hydrogen and discuss
your result.
8.9 Calculate by direct integration the expectation value of the radial position for each of the
Exercise 9.4.3. Ignore the fact that the hydrogen atom is a three-dimensional system and
pretend that
e?
H=
2m (R?)/2
(P² = P? + P; + P? , R² = X² + Y² +Z³)
corresponds to a one-dimensional problem. Assuming
ΔΡ- ΔR> h/2
estimate the ground-state energy.
Chapter 9 Solutions
Introduction To Quantum Mechanics
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- The three lowest energy levels of a hydrogen atom are -13.6 eV, -3.4 eV, and -1.5 eV. Assume that there is only one way to occupy any one of these levels. Calculate the relative probability that a hydrogen atom in thermal equilibrium in a star, at temperature T = 9674 K, is in its first excited state (at -3.4 eV) relative to its ground state (at -13.6 eV). Write your answer in exponential form. Recall that Boltzmann's constant can be written as 8.617 x 10-5 eV K-1.arrow_forward4.2 Evaluate a. A hydrogen atom is in a superposition state given by: 1 Y = [3V100-4210-2321] (29 b. The probability that the atom is found in each of the sub-states. The expectation value of the energy in this superposition state. Given R E = J = 72 n 13.6 n e Varrow_forwardProblem 3.36. Consider an Einstein solid for which both N and q are much greater than 1. Think of each oscillator as a separate "particle." (a) Show that the chemical potential is N+ - kT ln N (b) Discuss this result in the limits N > q and N « q, concentrating on the question of how much S increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?arrow_forward
- 1.7 A crystal has a basis of one atom per lattice point and a set of primitive translation vectors (in A): c = 1.5(i + j+ k), a = 3i, where i, j and k are unit vectors in the x, y and z directions of a Cartesian coordinate system. What is the Bravais lattice type of this crystal, and what are the Miller indices of the set of planes most densely populated with atoms? Calculate the volumes of the primitive unit cell and the conventional unit cell. b= 3j.arrow_forwardProblem 4.2 According to quantum mechanics, the electron cloud for a hydrogen atom in the ground state has a charge density. –2r/a p(r) = where q is the charge of the electron and a is the Bohr radius. Find the atomic polarizability of such an atom. [Hint: First calculate the electric field of the electron cloud, Ee(r); then expand the exponential, assuming r « a. For a more sophisticated approach, see W. A. Bowers, Am. J. Phys. 54, 347 (1986).]arrow_forwardFor Problem 9.18, how do I determine part A & B? This is from a chapter titled, "Electron Spin." This chapter is part of quantum mechanics.arrow_forward
- The three lowest energy levels of a hydrogen atom are -13.6 eV, -3.4 eV, and -1.5 eV. Place this atom in thermal contact with a reservoir and assume that there is only one way to occupy any one of these levels. Calculate the relative probability that this hydrogen atom at T = 316 K is in its first excited state (at -3.4 eV) relative to its ground state (at -13.6 eV). Write your answer in exponential form. An "eV" (electron volt) is the energy acquired by an electron accelerated across a 1 volt potential difference. This unit is used to describe electronic energy levels in atoms or solids (semiconductors, etc.). 1 eV = 1.602 x 10-19 J and Boltzmann's constant can be written as 8.617 x 10-5 eV K-1. If your calculator is unable to do this calculation try the web site https://www.wolframalpha.com In this site ex is entered as e^x, though exp(x) can also be used. If you haven't used this website before, a convenient tutorial can be found on youtube (for example,…arrow_forwardProblem 1: Estimate the probability that a hydrogen atom at room temperature is in one of its first excited states (relative to the probability of being in the ground state). Don't forget to take degeneracy into account. Then repeat the calculation for a hydrogen atom in the atmosphere of the star y UMa, whose surface temperature is approximately 9500K.arrow_forwardConsider the hydrogen atom. (a) For the ground state (n = 1), first excited states (n = 2), and second excited states (n = 3): 1. Calculate the energy En in eV. 2. Calculate all possible values of the angular momentum L and z-component angular momentum Lz (you can leave your values in terms of ¯h – e.g., L = 2¯h). 3. For each value of n and L, sketch the radial wave function and radial probability density. (b) Prove that, in the ground state, the electron is most likely to be found at the Bohr radius aB.arrow_forward
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